Integration Techniques Group Members Sam Taylor Patience Canty Austin Hood

Definition of an Integral - inverse of differentiation – Uses areas under curved surfaces centres of mass Volumes of solids Formulas for Integration IndefiniteDefinite

World Applications of Integration The Petronas Towers in Kuala Lumpur experience high forces due to winds. Integration was used to design the building for strength Historically, one of the first uses of integration was in finding the volumes of wine-casks (which have a curved surface) The Sydney Opera House is a very unusual design based on slices out of a ball. Many differential equations (one type of integration) were solved in the design of this building

World Applications of Integration cont. The Head Injury Criterion (HIC) The head Injury Criterion (HIC) was developed, it is based on the average value of the acceleration over the most critical part of the deceleration. It more accurately describes the likelihood of certain injuries in a crash The average value of the acceleration a(t) over the time interval t 1 to t 2 is given by For the HIC, this was modified (based on experimental data) as follows: The formula means: The HIC is the maximum value over the critical time period t sub 1 to t sub2 for the expression in {}. The index 2.5 is chosen for the head, based on experiments.

History of Integration Over 2000 years ago, Archimedes ( BC) found formulas for the surface areas and volumes of solids such as the sphere, the cone, and the paraboloid. His method of integration was remarkably modern considering that he did not have algebra, the function concept, or even the decimal representation of numbers. Leibniz ( ) and Newton ( ) independently discovered calculus. Their key idea was that differentiation and integration undo each other. Using this symbolic connection, they were able to solve an enormous number of important problems in mathematics, physics, and astronomy. newton-blogging-an-apple-tree-of-knowledge/

History of Integration cont. Gauss ( ) made the first table of integrals, and with many others continued to apply integrals in the mathematical and physical sciences. Cauchy ( ) took integrals to the complex domain. Riemann ( ) and Lebesgue ( ) put definite integration on a firm logical foundation. In the 20th century before computers, mathematicians developed the theory of integration and applied it to write tables of integrals and integral transforms.

History of Integration cont. In 1969 Risch made the major breakthrough in algorithmic indefinite integration when he published his work on the general theory and practice of integrating elementary functions. The capability for definite integration gained substantial power in Mathematica, first released in Comprehensiveness and accuracy have been given strong consideration in the development of Mathematica and have been successfully accomplished in its integration code

POWER RULE where C = Constant of integration u = Function n = Power du = Derivative = u = x du = dx n = 2 =

Examples 1. u = x du = dx n = 2 u = x du = dx n = 1 u = x du = dx n = o Solution:

2. u = 2x + 1 du = 2dx n = 15 Solution:

OR Solution:

u = 2x + 3 du = 2dx n = -1 Solution:

U - Substitution When to use U-Sub: The problem must be two algebraic functions One of them is NOT the derivative of the other Examples of this would include:

Examples 1. u = Solution:

2. Solution:

Integrating Powers of Sine and Cosine Rules and Ways to integrate: Integrating Odd Powers Integrating Odd and Even Powers Integrating Even Powers

Integrating Odd Powers Solution:

Integrating Odd and Even Powers Solution:

Integrating Even Powers Half – Angle Formulas Solution:

Integration by Parts If the functions are not related then use integration by parts Special things to Consider: Use lnx as the u variable. u = ln(x)

Example Solution:

Important/Unusual Integrals +C

Example 2 Solution:

Integration by Partial Fractions Must Haves: Expressions must be polynomials Power Rule must be used at some point Denominator is factorable, then partial factors Power or exponent = how many variables or fractions Example 1

Solution:

Example 2 Solution:

Definite Integration Used when the numerical bounds of the object are known First Fundamental Theorem of Calculus

Examples of Definite Problems Plug 2 in Plug 0 in Solution:

© Patience Canty, Austin Hood, and Sam Taylor Feb Bibliography l